Nonlinear quasi‐static finite element formulations for viscoelastic Euler–Bernoulli and Timoshenko beams

Abstract

Abstract Weak form finite element models for the nonlinear quasi‐static bending and extension of initially straight viscoelastic Euler–Bernoulli and Timoshenko beams are developed using the principle of virtual work. The mechanical properties of the beams are considered to be linear viscoelastic. However, large transverse displacements, moderate rotations and small strains are allowed by retaining the von Kármán strain components of the Green–Lagrange strain tensor in the formulation. The fully discretized finite element equations are developed using the trapezoidal rule in conjunction with a two‐point recurrence relation. The resulting finite element equations, therefore, necessitate data storage from the previous time step only, and not the entire deformation history. Membrane locking is eliminated from the Euler–Bernoulli formulation through the use of selective reduced Gauss–Legendre quadrature. Membrane and shear locking are both circumvented in the Timoshenko beam finite element by employing a family of high‐order Lagrange polynomials. A Newton–Raphson iterative scheme is used to solve the nonlinear finite element equations. Copyright © 2009 John Wiley & Sons, Ltd.